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4-1.Complex numbers
hard
Let $S=\left\{Z \in C: \bar{z}=i\left(z^2+\operatorname{Re}(\bar{z})\right)\right\}$. Then $\sum_{z \in S}|z|^2$ is equal to
A
$\frac{7}{2}$
B
$4$
C
$\frac{5}{2}$
D
$3$
(JEE MAIN-2023)
Solution
Let $Z=x+$ iy, $x \in R, y \in R$
$x-i y=i\left(x^2-y^2+(2 x y) i+x\right)$
$x =- 2 x x$
$- y =- y ^2+ x ^2+ x$
$\Rightarrow x=0, y=-\frac{1}{2}(\text { from }(1))$
If $x \neq 0$, then $y =0,1$
If $y =-\frac{1}{2}$, then $x =\frac{1}{2},-\frac{3}{2}$
$Z =0+ i 0,0+ i , \frac{1}{2}-\frac{ i }{2},-\frac{3}{2}-\frac{ i }{2}$
Standard 11
Mathematics
